Proceedings of the European Cognitive Science Conference, Delphi, Greece, May 2007, pp. 312–317
A Generalization of Hebbian Learning in Perceptual and Conceptual Categorization
Harry E. Foundalis (harry@cogsci.indiana.edu)
Center for Research on Concepts and Cognition, Indiana University
Bloomington, IN 47408, USA
Maricarmen Martínez (m.martinez97@uniandes.edu.co)
Department of Mathematics, Universidad de los Andes
Bogotá, Colombia
Abstract
An algorithm for unsupervised competitive learning is
presented that, at first sight, appears as a straightforward
implementation of Hebbian learning. The algorithm is then
generalized in a way that preserves its basic properties but
facilitates its use in completely different domains, thus
rendering Hebbian learning a special case of its range of
applications. The algorithm is not a neural network
application: it works not at the neural but at the conceptual
level, although it borrows ideas from neural networks. Its
performance and effectiveness are briefly examined.
Introduction
Traditionally, learning has been considered to be one of the
foundational pillars of cognition. In 1949, Donald Hebb
expressed the basic idea of association learning as follows:
when two neurons are physically close and are repeatedly
activated together, some chemical changes must occur in
their structures that signify the fact that the two neurons
fired together (Hebb, 1949). Neuroscientists and cognitive
science researchers dubbed this idea “Hebbian learning”,
and took it to mean that whenever two percepts appear
together, we (or animals in general) learn an association
between them. The “how” is usually left to algorithms in
artificial neural networks (ANN’s). James McClelland
points out in a recent study that the potential of Hebbian
learning has been largely underestimated (McClelland,
2006). McClelland focuses on the neuronal level, but refers
also to work at the conceptual level (the focus of the present
article); specifically, in categorization (e.g., Schyns, 1991)
and in self-organizing topological maps (Kohonen, 1982).
In what follows, an algorithm that uses Hebbian learning
at the conceptual level is introduced through an example
that appears in human cognition. It is then generalized in a
way that, while retaining its basic properties, makes it
applicable to problems that involve the detection (and hence
the learning) of sets of entities that are most closely related
to each other.
The Basic Algorithm
Suppose we are given input consisting of a pair of elements,
i.e., a drawing depicting some familiar objects, and a phrase
with identifiable words, the meaning of which is unknown
(Figure 1). The phrase is supposed to be “about” the objects,
their properties, and/or relations. The problem is to find the
correct associations between words and percepts; or, stated
otherwise, to learn the meaning of the words (in some
rudimentary sense of the word “meaning”1).
nae triogon aems es nae cycol
Figure 1: A drawing, and its associated phrase
The drawing in Figure 1 shows two familiar geometric
objects, and the phrase is given underneath the objects in
italics. The learner is not expected to know any words for
these objects or their properties in some other language;
indeed, the phrase might originate from what will turn out to
be the learner’s native language.2 The learner will receive a
number of such examples, pairing visual and linguistic data,
and the algorithm described below will discover the correct
associations between words and visual percepts. This
problem has been examined also by Deb Roy (Roy, 2002),
but Roy’s learning succeeds by batch processing, whereas
the learning described in the present article is incremental.
Suppose the learner is capable of perceiving some objects,
features, and/or relations — collectively called percepts —
by looking at the visual input, which activate corresponding
concepts in long-term memory (LTM). Also, the learner can
identify (i.e., separate from each other) the words in the
given phrase.3 In reality, an infant learning a language will
not identify all the words in a phrase, but it does not harm
the generality of our algorithm to assume so.
As a first — perhaps naïve — step in our effort to
discover which word corresponds to which percept, let us
adopt the following simplistic strategy: associate every
concept with every word. Figure 2 depicts this idea. LTM
concepts that were activated by percepts in the input are
listed on the top row, and the words of the given phrase are
added to LTM and shown on the bottom row. Notice that
1
Contrary to our simplifying assumption, in reality words and
percepts are not associated according to a 1–1 correspondence.
2
However, infants do not become native speakers by hearing
phrases that refer to geometric objects. What is described here is an
abstraction of a real-world situation.
3
Thus, suppose the “word-segmentation problem” is solved.
Proceedings of the European Cognitive Science Conference, Delphi, Greece, May 2007, pp. 312–317
each word is shown only once (for instance, “nae”, which is
repeated in the input, is not repeated in Figure 2).
“in”
nae
triogon
3
aems
1
es
cycol
Figure 2: First uninformed step in the algorithm
The top row of nodes in Figure 2 shows only some of the
LTM concepts that could be activated by the input;
specifically, “circle”, “triangle”, “in”, “three-ness”, “right
angle”, and “one-ness”. A look by a different agent (or by
the same agent at a different time) might activate different
concepts, such as “round”, “surrounds”, “slanted line”, etc.
Those shown in Figure 2 are the ones that happened to be
activated most strongly in the given learning session.
Next, the learning session continues with a second pair of
visual and linguistic input.
and U – “nae”) have their associations reinforced (shown in
Figure 4 with bold lines), whereas the associations of those
pairs of input 1 that did not co-occur in input 2 have faded
slightly (shown in Figure 4 with dotted lines). At the same
time, as before, the added words and concepts form all
possible associations (lines of normal thickness).
The expectation is now clear: given more instances of
paired visual and linguistic input (not too many — at most a
few dozen) the “correct” associations (i.e., those intended by
the input provider) should prevail, while those that are
“noise” (unintended) should be eliminated, having faded
beyond some detection threshold.
“in”
nae
triogon
aems
3
1
es
cycol
2
doy
ot
Figure 5: Desired (final) set of associations
Figure 5 shows an approximation of the desired result of
this procedure. (There will be many more concepts and
words introduced on the two rows of nodes, and most, but
not all associations will be “correct”: there will be spurious
associations that did not fade to the point of elimination.)
Activations of Associations
Figure 3: A second pair of visual and linguistic input
The image in Figure 3 activates again some concepts in
LTM, some of which were activated also by the input of
Figure 1 (e.g., “triangle”). Linguistically, there are some old
and new words. Also, one word looks very much like an old
one (“triagon” vs. “triogon”). A suitable word-identification
algorithm would not only see the similarity, but also analyze
their morphological difference, which would later allow the
agent to make an association between this morphological
change and the visual percept that was responsible for the
change. But at this stage no handling of morphology is
required by the algorithm; suffice it to assume that the two
words, “triagon” and “triogon”, are treated as “the same”.
The algorithm now adds the new concepts to the list at the
top (that is, it forms the union of the sets of the old and new
concepts) and the new words to the list at the bottom.
“in”
nae
triogon
aems
3
1
es
cycol
2
doy
ot
Figure 4: Some associations are reinforced
For lack of horizontal space, only a few of the new words
and concepts are shown in Figure 4. The important
development in Figure 4 is that those concept – word pairs
that are repeated in inputs 1 and 2 (such as U – “triogon”
For the above-outlined algorithm to converge to the desired
associations, a precise mechanism of association fading and
reinforcement must be established. If the associations fade
too fast, for example, they will all be eliminated before
additional evidence arrives making them strong enough to
survive to the end of the process; if they fade too slow, all
associations (including “noise”) will eventually survive;
finally, if a minute degree of fading is allowed even after the
system receives no further input (waiting, doing nothing),
then given enough time all associations will drop back to
zero and the system will become amnesic. To counter these
problems, the “strength” of each association, which we call
activation, is expected to have the following properties.
1
α (x)
discrete
steps
activation
doy triagon, ot nae ainei scelisoes
0
x
1
Figure 6: How activation changes over time
The activation of an association is the quantity α(x) on the
y-axis of Figure 6. Suppose that, at any point in time, there
is a quantity x on the x-axis ranging between 0 and 1, but
that can take on values only along one of the discrete points
shown on the x-axis. (The total number of these points is an
Proceedings of the European Cognitive Science Conference, Delphi, Greece, May 2007, pp. 312–317
Implementation
The above-described principles were implemented as part of
Phaeaco, a visual pattern-recognition system (Foundalis,
2006), in which the visual input consisted of a 200 × 200
rectangle of black-and-white dots. A “training set” consisted
of 50 pairs of the form [image, phrase], where the images
were similar to those shown earlier in Figure 1 and Figure 3
(they contained geometric figures), and phrases (in English)
were likewise relevant to the content of their paired images.
Phaeaco is capable of perceiving the geometric structure of
such figures, building an internal representation of the
structure in its “working memory” according to its
architectural principles, and letting the parts of this
representation activate the corresponding concepts in its
long-term memory (LTM). Thus, if a square is drawn in the
input, the following concepts might be activated in
Phaeaco’s LTM: square, four, four lines, parallel lines,
equal lengths, interior, four vertices, right angle, equal
angles, etc. (listed in no particular order). Some of these
concepts will be more strongly activated than others, due to
the principles in Phaeaco’s architecture (e.g., square is more
“important” than vertex, because a vertex is only a part of
the representation of a square). Normally, Phaeaco starts
with some “primitives”, or a priori known concepts (such as
point, line, angle, interior, etc.), and is capable of learning
other concepts (such as square) based on the primitives.4
However, for the purposes of the described experiment we
suppressed Phaeaco’s mechanism of learning new concepts,
so as to avoid interference with the learning of associations
between words and concepts. Thus, we worked with an
LTM that already included composite concepts such as
square, triangle, etc. — i.e., anything that might appear in
the visual input of a training set.
An additional simplifying assumption was that the
rudimentary morphology of English was ignored. Thus,
words such as “triangle” and “triangles” were treated as
identical; so were all forms of the verb “to be”, etc.
The entire training set can be found in Martínez (2006).
Performance
The following graph (Figure 7) presents the progress of the
learning of correct associations over time.
correct assoc.
important parameter, to be discussed soon.) Thus, x makes
only discrete steps along the x-axis. Then the quantity α (x)
is given by the sigmoid-like function α, also ranging
between 0 and 1. Below, we explain how and when x moves
along the x-axis, and why α must be sigmoid-like.
Activations fade automatically, as time goes by. This
means that if a sufficient amount of time elapses, x makes
one discrete step backwards in its range. Consequently, α (x)
is decreased, since α is monotonic. “Time” is usually
simulated in computational implementations, but in a realtime learning system it is assumed to be the familiar
temporal dimension (of physics).
Activations are reinforced by the learning system by
letting x make a discrete step forward in its range. Again,
the monotonicity of α implies that α (x) is increased.
However, when x exceeds a threshold that is just before the
maximum value 1, the number of discrete steps along the xaxis is increased somewhat (the resolution of segmenting
the x-axis grows). This implies that the subsequent fading of
the activation will become slower, because x will have more
backward steps to traverse along the x-axis. The meaning of
this change is that associations that are well established
should become progressively harder to fade, after repeated
confirmations of their correctness. The amount by which the
number of steps is increased is a parameter of the system.
Finally, an explanation must be given for why α must be
sigmoid-like. First, observe that α must be increasing strictly
monotonically, otherwise the motion of x along the x-axis
would not move α (x) in the proper direction. Now, of all the
monotonically increasing curves that connect the lower-left
and upper-right corners of the square in Figure 6, the
sigmoid is the most appropriate shape for the following
reasons: the curve must be initially increasing slowly, so that
an initial number of reinforcements starting from x = 0 does
not result in an abrupt increase in α (x). This is necessary
because if a wrong association is made, we do not want a
small number of initial reinforcements to result in a
significant α (x) — we do not wish “noise” to be taken
seriously. Conversely, if x has approached 1, and thus α (x)
is also close to 1, we do not want α (x) to suddenly drop to
lower values; α must be conservative, meaning that once a
significant α (x) has been established it should not be too
easy to “forget” it. This explains the slow increase in the
final part of the curve in Figure 6. Having established that
the initial and final parts must be increasing slowly, there
are only few possibilities for the middle part of a monotonic
curve, hence the sigmoid shape of curve α.
12
10
8
6
4
2
0
0
5 10 15 20 25 30 35 40 45 50
time
Figure 7: Progress of learning over time
The graph in Figure 7 shows that the average number of
correctly learned associations (y-axis) is a generally
increasing function with respect to the number of input pairs
presented (x-axis). Since the input pairs were presented at
regular intervals in our implementation, the x-axis can also
be seen as representing time. The y-values are averages of a
large number (100) of random presentation orderings of the
training set. The gradient of the slope of the curve (the
“speed of learning”) depends on a variety of factors,
including the settings of the activation parameters and the
content of the visual and linguistic input. Under most
reasonable settings, however, the learning curve appears to
slowly increase with respect to time, and this was the main
objective of our implementation. In addition, notice that our
algorithm requires only one presentation of the training set,
as opposed to multiple epochs typically required in ANN’s.
4
This learning ability is completely unrelated to the algorithm of
learning associations, discussed in the present text.
Proceedings of the European Cognitive Science Conference, Delphi, Greece, May 2007, pp. 312–317
A Seemingly Different Application
The above-described process can be generalized in an
obvious — and rather trivial — way, to any case in which
data from two different sets can be paired, and associations
between their members can be discerned and learned. For
example, consider discovering the cause for an allergy. One
set is “the set of all possible pathogens that could cause me
this allergy”, and the second set has a single member, the
event (or fact) “I have this allergy”. What is needed is an
association of the form “pathogen X causes this allergy to
me”. We might observe the candidate causes during a long
period of time, and subconsciously only, without actively
trying to discover the cause. Over time, some candidates are
reinforced due to repetition, whereas others fade. Given the
right conditions (sufficient number of repetitions and not too
long an interval of observation time forcing all associations
to fade back to zero), we might reach an “Aha!” moment, in
which one of the associations becomes strong enough to be
noticed consciously.5 Similarly, some examples of scientific
discovery (“what could be the cause of phenomenon X?”)
can be implemented algorithmically by means of the same
process. However, the generalization discussed in what
follows goes beyond the pairing of elements of two sets.
Suppose that the input is in the visual modality only, and
consists of the shapes shown in Figure 8 (a).
(a)
(b)
Figure 8 (a): Sample visual input; (b): a few pixels seen
Suppose also that a visual processing system examines
the black pixels of this input, not in some systematic, top-tobottom, left-to-right fashion, but randomly. Indeed, this is
the way in which Phaeaco examines its input (Foundalis,
2006). After some fast initial processing, Phaeaco has seen a
few pixels that belong to the central (“median”) region of
the parallelogram and/or the circle, as shown in Figure 8 (b).
Shortly afterwards, a few more pixels become known, as
shown in Figure 9 (a), where the outlines of the figures are
barely discernible (to the human eye). Within similarly short
time, enough pixels have been seen — as in Figure 9 (b) —
for the outlines to become quite clear, at least to the human
eye. What is needed is an algorithm that, when employed by
the visual processing system, will make it as capable and
fast in discerning shapes from a few pixels as the human
visual system. To achieve this, Phaeaco employs the
following algorithm.
As soon as the first pixels become known (Figure 8 (b)),
Phaeaco starts forming hypotheses about the line segments
on which the so-far known pixels might lie (Figure 10).
Legend:
weaker
activation
stronger
activation
Figure 10: First attempts to “connect the dots” with lines
Most of these hypotheses will turn out to be wrong (“false
positives”). But it does not matter. Any of these initial
hypotheses that are mere “noise” will not be reinforced, so
their activations will fade over time; whereas the correct
hypotheses for line segments will endure. Thus, Phaeaco
entertains line-segment detectors, which are line segments
equipped with an activation value. The method of least
squares is used to fit points to line segments, and the
activation value of each detector depends both on how many
points (pixels) participate in it, and how well the points fit.
Note that only one of the detectors in Figure 10 is a “real”
one, i.e., one that will become stronger and survive until the
end (the nearly horizontal one at the top); but Phaeaco does
not know this yet.
Subsequently more points arrive, as in Figure 9 (a). Some
of the early detectors of Figure 10 will receive one or two
more spurious points, but their activations will fade more
than they will be reinforced. Also, a few more spurious
detectors might form. But, simultaneously, the “real” ones
will start emerging (Figure 11).
Legend:
weak
medium
strong
(a)
(b)
Figure 9 (a) – (b): More pixels seen
5
It might be incorrect to make this association, thus concluding the
wrong cause, but it is the process that concerns us here.
Figure 11: More points keep the detectors evolving
Note that several of the early detectors in Figure 10 have
disappeared in Figure 11, because they were not fed with
more data points and “died”. Thus, the fittest detectors
survive at the expense of other detectors that do not explain
Proceedings of the European Cognitive Science Conference, Delphi, Greece, May 2007, pp. 312–317
the data sufficiently well. The last (but not final) stage in
this series is shown in Figure 12, below.
Legend:
fade, and we end up with a clearer view of the correct
categories in this space, as shown in Figure 14. (The space
dimensionality has been kept constant, equal to 2, for
purposes of illustration.)
weak
medium
strong
Figure 12: Most of the surviving detectors are “real”
In Figure 12, the desired detectors that form the sides of
the parallelogram are among the survivors. Also, the tiny
detectors that will be used later to identify the circle have
started appearing. In the end, all “true” detectors will be
reinforced with more points and deemed the line segments
of the image, whereas all false detectors will fade and die.
The particular reasons why Phaeaco employs this rather
unconventional (randomized) processing of its visual input
are explained in Foundalis (2006). For the purposes of the
present article it is important to point out that the above
procedure is extendible to any case where the detection of
the most salient among a group of entities (objects, features,
etc.) is required. The reason why this process is similar to
the Hebbian association-building will be discussed soon.
Figure 14: More points help to better discern the categories
The algorithm is described in detail in Foundalis (2006,
pp. 228–235). In Phaeaco, categories formed in this way
(see Figure 15) are defined by the barycenter (“centroid”) of
the set, its standard deviation, and a few more statistics.
They also include a few of the initially encountered data
points. Thus, they are an amalgam of the prototype (Rosch,
1975; Rosch and Mervis, 1975) and exemplar (Medin and
Schaffer, 1978; Nosofsky, 1992) theories of category
representation, following the principles of the FARG family
of cognitive architectures (Hofstadter, 1995).
Generalizing to Categorization
The process described in the previous section is a specific
application of the more general and fundamental process of
categorization. An example will suffice to make this clear.
Suppose we are visitors at a new location on Earth where
we observe the inhabitants’ faces. Initially all faces appear
unfamiliar, and, having seen only a few of them, we can do
no better than place them all in one large category,
“inhabitant of this new place”, as abstracted in Figure 13.
Figure 13: Categorization in abstract space of facial types
Note that the face-space on the left side of Figure 13
includes only two facial dimensions, x and y, since each dot
represents a face. In reality the space is multidimensional,
and we become capable of perceiving more dimensions as
our experience is enriched with examples. An initial stage at
categorization is shown on the right side of Figure 13. The
dashed outlined regions are detectors of categories, almost
identical in nature with the detectors of lines of the previous
section. As before, new examples are assigned to the group
in which they best fit, statistically. Over time, as more
examples become available, the categories that are “noise”
Figure 15: The categories finally become clear
The connection with the earlier discussion of line segment
detection should now be clear: the same mechanism can be
employed in both cases for the gradual emergence and
discerning of the categories, or concepts, in the given space.
The two cases, perceptual detection of lines and conceptual
detection of categories, are very nearly isomorphic in
nature. In general, the discerning of concepts (of any kind,
whether concrete objects or abstract categories), might be
thought to rely on a process similar to the one described
here. For concrete objects, this process has been completely
automated and has been hardwired by evolution, so that
very few parameters of it need be learned; for other, more
abstract and gradually discerned categories, it might look
more like the mechanism outlined above.
Discussion
Two seemingly different case-studies were discussed in the
previous sections: a Hebbian-like “association-building by
co-occurrences” process, and one of gradual discerning of
categories in a multidimensional space. What is the deeper
commonality between the two?
Both are about discerning entities. In the case of Hebbian
learning the entities are the associations, whereas in the
Proceedings of the European Cognitive Science Conference, Delphi, Greece, May 2007, pp. 312–317
other examples mentioned the entities are categories that
appear as clusters of objects. The discerning is gradual, and
initially there is a lot of “noise” in the form of wrongly
detected entities. But over time the “noise” fades away and
the correct entities emerge highly activated.
This is an unsupervised learning algorithm, the success of
which depends on the appropriate settings of the parameters
of the activations of entities. Activations (illustrated in
Figure 6) must have the following properties:
• The initial increase in activation must be conservative,
to avoid an early commitment to “noise”.
• The activation strength must be fading automatically
as time goes by, to avoid having all activations of
associations or categories eventually reach the value 1.
• Those associations or categories that appear to stand
consistently above others in strength must curb the
fading speed of their activations, to avoid forgetting
even the most important ones among them over time.
It might be thought that our choice of setting the starting
value of an activation to 0 results in a strictly deterministic
process. However, this criticism is superficial. Although our
model is indeed deterministic, in a real-world situation the
order in which the input is encountered is practically never
predetermined. Indeterminism arises from the real world.
Thus, what is required is that any (random) order allows our
algorithm to run, and indeed, in our measurements we
varied the input presentation order randomly, observing no
dependence of the algorithm on any particular input order.
Although ideas similar to the above have traditionally
been viewed as belonging to the neural level that inspired
ANN’s, our work supports the idea that it is possible that the
same principles have been utilized by human cognition at a
higher conceptual level. This is not without precedent in
material evolution and has been noted also in other scientific
disciplines (e.g., physics, chemistry, biology). For example:
• The structure of a nucleus with surrounding material is
found in atoms at the quantum level, and in planetary
systems and galaxies at the macroscopic level. It is
also found in biology in eukaryotic cells, and in
animal societies organized around a leading group,
with “distances” of individuals from the leader, or
leaders, varying according to their social status.
• The notion of “force”: in the quantum world, forces
are interactions of fermions through the exchange of
bosons (e.g., Ford, 2004). Chemically, forces are
responsible for molecular structure. In biology, a force
is exerted usually by a muscular structure. By analogy,
a “force” can be of psychological or social nature.
• The notion of “wave”: in the microworld there are
waves of matter, or waves of probability; in the
macroworld there are waves of sound, fluids, gravity,
etc. More abstractly, there are “waves” of fashion,
cultural ideas, economic crises, etc.
In a similar manner, we suggest that, through evolutionary
mechanisms, cognition abstracted from what was initially
employed as a simple association-building mechanism in
creatures that appeared early on in evolutionary history to a
conceptual categorization method, which finds its most
versatile expression and application in human cognition.
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